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 Probl. Peredachi Inf., 2008, Volume 44, Issue 2, Pages 75–95 (Mi ppi1272)

Large Systems

Exact Asymptotics of Small Deviations for a Stationary Ornstein–Uhlenbeck Process and Some Gaussian Diffusion Processes in the $L_p$-Norm, $2\le p\le\infty$

V. R. Fatalov

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: We prove results on exact asymptotics of the probabilities
$$\mathrm{P}\{\int_0^1|\eta(t)|^p dt\leq\varepsilon^p\},\quad\varepsilon\to 0,$$
where $2\leq p\leq\infty$, for two types of Gaussian processes $\eta(t)$, namely, a stationary Ornstein–Uhlenbeck process and a Gaussian diffusion process satisfying the stochastic differential equation
Z(0)=0. \end{gather*}
Derivation of the results is based on the principle of comparison with a Wiener process and Girsanov's absolute continuity theorem.

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English version:
Problems of Information Transmission, 2008, 44:2, 138–155

Bibliographic databases:

UDC: 621.391.1:519.2

Citation: V. R. Fatalov, “Exact Asymptotics of Small Deviations for a Stationary Ornstein–Uhlenbeck Process and Some Gaussian Diffusion Processes in the $L_p$-Norm, $2\le p\le\infty$”, Probl. Peredachi Inf., 44:2 (2008), 75–95; Problems Inform. Transmission, 44:2 (2008), 138–155

Citation in format AMSBIB
\Bibitem{Fat08} \by V.~R.~Fatalov \paper Exact Asymptotics of Small Deviations for a~Stationary Ornstein--Uhlenbeck Process and Some Gaussian Diffusion Processes in the $L_p$-Norm, $2\le p\le\infty$ \jour Probl. Peredachi Inf. \yr 2008 \vol 44 \issue 2 \pages 75--95 \mathnet{http://mi.mathnet.ru/ppi1272} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2435241} \elib{http://elibrary.ru/item.asp?id=13583794} \transl \jour Problems Inform. Transmission \yr 2008 \vol 44 \issue 2 \pages 138--155 \crossref{https://doi.org/10.1134/S0032946008020063} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000257584100006} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-48249091277} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. V. R. Fatalov, “Small deviations for two classes of Gaussian stationary processes and $L^p$-functionals, $0<p\le\infty$”, Problems Inform. Transmission, 46:1 (2010), 62–85
2. Ya. Yu. Nikitin, R. S. Pusev, “The exact asymptotic of small deviations for a series of Brownian functionals”, Theory Probab. Appl., 57:1 (2013), 60–81
3. V. R. Fatalov, “Asymptotic behavior of small deviations for Bogoliubov's Gaussian measure in the $L^p$ norm, $2\le p\le\infty$”, Theoret. and Math. Phys., 173:3 (2012), 1720–1733
4. V. R. Fatalov, “Ergodic means for large values of $T$ and exact asymptotics of small deviations for a multi-dimensional Wiener process”, Izv. Math., 77:6 (2013), 1224–1259
5. V. R. Fatalov, “Gaussian Ornstein–Uhlenbeck and Bogoliubov processes: asymptotics of small deviations for $L^p$-functionals, $0<p<\infty$”, Problems Inform. Transmission, 50:4 (2014), 371–389
6. V. R. Fatalov, “Weighted $L^p$, $p\ge2$, for a wiener process: Exact asymptoties of small deviations”, Moscow University Mathematics Bulletin, 70:2 (2015), 68–73
7. Lototsky S.V., “Small Ball Probabilities For the Infinite-Dimensional Ornstein-Uhlenbeck Process in Sobolev Spaces”, Stoch. Partial Differ. Equ.-Anal. Comput., 5:2 (2017), 192–219
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